Let us consider the boundary value problem (BVP) generated by the Sturm-Liouville equation

and the boundary condition

where

is a real-valued function and

is a spectral parameter. The bounded solution of (1.1) satisfying the condition

will be denoted by

. The solution

satisfies the integral equation

It has been shown that, under the condition

the solution

has the integral representation

where the function

is defined by

. The function

is analytic with respect to

in

, continuous

, and

holds [1, chapter 3].

The functions

and

are called Jost solution and Jost function of the BVP (1.1) and (1.2), respectively. These functions play an important role in the solution of inverse problems of the quantum scattering theory [

1–

4]. In particular, the scattering date of the BVP (1.1) and (1.2) is defined in terms of Jost solution and Jost function. Let

,

, be the zeros of the Jost function, numbered in the order of increase of their moduli (

) and

are bounded solutions of the BVP (1.1) and (1.2), where

is the scattering function [

1–

4]. Using (1.7), we get that

hold. The collection of quantities
that specify to as the behaviour of the radial wave functions
and
at infinity is called the scattering of the BVP (1.1) and (1.2).

Let us consider the self-adjoint system of differential equations of first order

where
and
are real-valued continuous functions. In the case
,
, where
is a potential function and
the mass of a particle, (1.11) is called stationary Dirac system in relativistic quantum theory [5, chapter 7]. Jost solution and the scattering theory of (1.11) have been investigated in [6].

Jost solutions of quadratic pencil of Schrödinger, Klein-Gordon, and
-Sturm-Liouville equations have been obtained in [7–9]. In [10–17], using the analytical properties of Jost functions, the spectral analysis of differential and difference equations has been investigated.

Discrete boundary value problems have been intensively studied in the last decade. The modelling of certain linear and nonlinear problems from economics, optimal control theory, and other areas of study has led to the rapid development of the theory of difference equations. Also the spectral analysis of the difference equations has been treated by various authors in connection with the classical moment problem (see the monographs of Agarwal [18], Agarwal and Wong [19], Kelley and Peterson [20], and the references therein). The spectral theory of the difference equations has also been applied to the solution of classes of nonlinear discrete Korteveg-de Vries equations and Toda lattices [21].

Now let us consider the discrete Dirac system

with the boundary condition

where

is the forward difference operator:

and

is the backward difference operator:

;

and

are real sequences. It is evident that (1.12) is the discrete analogy of (1.11). Let

denote the operator generated in the Hilbert space

by the BVP (1.12) and (1.13). The operator

is self-adjoint, that is,

. In the following, we will assume that, the real sequences

and

satisfy

In this paper, we find Jost solution of (1.12) and investigate analytical properties and asymptotic behaviour of the Jost solution. We also show that,
, where
denotes the continuous spectrum of
, generated in
by (1.12) and (1.13).

We also prove that under the condition (1.14) the operator
has a finite number of simple real eigenvalues.